Talk:Hypergeometric function

Pochhammer symbol
I'm not familiar to this topic, but according to this article, (q)n is defined by:


 * $$(q)_n =

\begin{cases} 1                                               & \mbox{if } n = 0 \\ q \left( q-1 \right) \cdots \left( q-n+1 \right) & \mbox{if } n > 0 \end{cases} $$

Jdh8 (talk) 15:16, 6 December 2012 (UTC)


 * Looks like a mistake. It should be the rising factorial $$q^{(n)}$$. This would be consistent with the definition given in the article about generalized hypergeometric function. — Preceding unsigned comment added by 141.89.116.54 (talk) 12:48, 14 February 2013 (UTC)


 * Yeah, don't confuse rising and falling; they're different. There are several different conventions and notations in common use, and you need to take care not to get tripped up. All of the WP articles in this area were all self-consistent with one another, last time I looked.  User:Linas (talk) 06:29, 1 December 2013 (UTC)


 * Ok, but the problem is that Wikipedia is not consistent with itself. Just check the reference to Pochhammer symbol Jdh8 gives. 2001:718:2:1634:224:1DFF:FE13:8CBC (talk) 14:19, 26 June 2014 (UTC)

No valid antecedent
"It is a solution of a second-order..." does not have a valid antecedent. — Preceding unsigned comment added by 98.67.106.148 (talk) 06:30, 22 September 2013 (UTC)

Identities?
As I remember it, research into the discovery of and classification of identities remains an active topic; many algorithms have been found that can generate classes of identities, but there is no complete categorization of them all -- i.e. there is no known algo that can generate all identites. This is one of the more peculair and modern aspects of this beast, I'm surprised its not covered. User:Linas (talk) 06:27, 1 December 2013 (UTC)

Did something get lost in this edit?
Something lost? I get that feeling, but do not know. YohanN7 (talk) 16:22, 12 October 2014 (UTC)


 * Nope! There were two edits that should be taken as one. My mistake. YohanN7 (talk) 16:28, 12 October 2014 (UTC)

higher order transformations
Vidunas classified 2F1 --> 2F1 transformations. I added one example to the main page (the entry 1/2,1/3,1/8 --> 1/4,1/8,1/8 in Table 2 in his paper). To guard against typing errors, here is the example in computer-readable form so one can check it with a Maple computation:
 * hypergeom([1/4, 3/8], [7/8], z ) * (z^4-60*z^3+134*z^2-60*z+1)^(1/16) = hypergeom([1/48, 17/48], [7/8], (-432*z*(z-1)^2*(z+1)^8 / (z^4-60*z^3+134*z^2-60*z+1)^3) );
 * series(lhs(%)-rhs(%), z=0, 30); MvH (talk) 14:00, 15 April 2015 (UTC)MvH

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Exponents
The term "exponents" is used 4 times on this page, but never defined.

1.136.107.136 (talk) 13:57, 21 October 2018 (UTC)

Discussion
The formula in the chapter (the hypergeometric series) consisting of the limit of the division taken on negative integers is wrong because the function gamma take infinite values on negative integers so the limit is 0. — Preceding unsigned comment added by Boutarfa Nafia (talk • contribs) 17:52, 1 March 2022 (UTC)


 * The formula is not wrong – note that the numerator approaches infinity. A1E6 (talk) 22:16, 11 April 2022 (UTC)

Discussion
There is a profound error in the conventions of hypergeometric series definition because it is the rising factorial which should be used in the definition not the descending factorial otherwise it is wrong. — Preceding unsigned comment added by Boutarfa Nafia (talk • contribs) 14:37, 8 April 2022 (UTC)


 * Here, $$(q)_n$$ is the rising factorial. It's even written right next to it in the article. Also note that some authors use different notation. A1E6 (talk) 22:12, 11 April 2022 (UTC)

Why are a and b separated by a comma, but c and z are separated by a semicolon?
Is this a sign of something fundamental? Lockywolf (talk) 02:22, 22 August 2023 (UTC)


 * As I understand, the semicolon before 'c' puts the (c)_n coefficients in the denominators of series representation; the semicolon before 'z' denotes z is the argument. It makes more sence when considering the generalized hypergeometric functions. 85.143.106.70 (talk) 12:01, 3 October 2023 (UTC)

Asymptotics
I think we need to add the formula relating 2F1 hypergeometric functions of 'z' and of '1/z'. It is the most convenient way of calculating the large argument asymptotics, and there are no other asymptotic formulas presented in the wikipage. 85.143.106.70 (talk) 12:05, 3 October 2023 (UTC)